Let $f$ be a differentiable function with $f ( 4 ) = 3$. On the interval $0 \leq x \leq 7$, the graph of $f ^ { \prime }$, the derivative of $f$, consists of a semicircle and two line segments, as shown in the figure above. (a) Find $f ( 0 )$ and $f ( 5 )$. (b) Find the $x$-coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$. Justify your answer. (c) Let $g$ be the function defined by $g ( x ) = f ( x ) - x$. On what intervals, if any, is $g$ decreasing for $0 \leq x \leq 7$ ? Show the analysis that leads to your answer. (d) For the function $g$ defined in part (c), find the absolute minimum value on the interval $0 \leq x \leq 7$. Justify your answer.
Let $f$ be a differentiable function with $f ( 4 ) = 3$. On the interval $0 \leq x \leq 7$, the graph of $f ^ { \prime }$, the derivative of $f$, consists of a semicircle and two line segments, as shown in the figure above.\\
(a) Find $f ( 0 )$ and $f ( 5 )$.\\
(b) Find the $x$-coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$. Justify your answer.\\
(c) Let $g$ be the function defined by $g ( x ) = f ( x ) - x$. On what intervals, if any, is $g$ decreasing for $0 \leq x \leq 7$ ? Show the analysis that leads to your answer.\\
(d) For the function $g$ defined in part (c), find the absolute minimum value on the interval $0 \leq x \leq 7$. Justify your answer.